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Planar Drawings of Plane Graphs
, 2000
"... this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results. ..."
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Cited by 13 (3 self)
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this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results.
Schnyder woods and orthogonal surfaces
 In Proceedings of Graph Drawing
, 2006
"... In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain connections between these seemingly unrelated ..."
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Cited by 7 (3 self)
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In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain connections between these seemingly unrelated notions. We use these connections for an intuitive proof of the BrightwellTrotter Theorem which says, that the face lattice of a 3polytope minus one face has order dimension three. Our proof yields a linear time algorithm for the construction of the three linear orders that realize the face lattice. Coplanar orthogonal surfaces are in correspondence with a large class of convex straight line drawings of 3connected planar graphs. We show that Schnyder’s face counting approach with weighted faces can be used to construct all coplanar orthogonal surfaces and hence the corresponding drawings. Appropriate weights are computable in linear time. 1
Orthogonal Drawings Based On The Stratification Of Planar Graphs
, 2000
"... Several algorithms have been proposed to draw planar graphs using 2visibility and Kandinsky Models. Here, we propose three new algorithms implementing these models in linear time using small grid sizes and few bends. These algorithms are all based on the construction of a particular layered spannin ..."
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Cited by 4 (2 self)
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Several algorithms have been proposed to draw planar graphs using 2visibility and Kandinsky Models. Here, we propose three new algorithms implementing these models in linear time using small grid sizes and few bends. These algorithms are all based on the construction of a particular layered spanning tree called Stratification. A linear time algorithm that computes a stratification is also presented.
Orthogonal Surfaces and their CPorders
, 2007
"... Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich structure with c ..."
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Cited by 2 (2 self)
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Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial already the three dimensional case has a rich structure with connections to Schnyder woods, planar graphs and 3polytopes. Our objective is to detect more of the structure of orthogonal surfaces in four and higher dimensions. In particular we are driven by the question which nongeneric orthogonal surfaces have a polytopal structure. We review the state of knowledge of the 3dimensional situation. On that basis we introduce terminology for higher dimensional orthogonal surfaces and continue with the study of characteristic points and the cporders of orthogonal surfaces, i.e., the dominance orders on the characteristic points. In the generic case these orders are (almost) face lattices of polytopes. Examples show that in general cporders can lack key properties of face lattices. We investigate extra requirements which may help to have cporders which are face lattices. Finally, we turn the focus and ask for the realizability of polytopes on orthogonal surfaces. There are criteria which prevent large classes of simplicial polytopes from being realizable. On the other hand we identify some families of polytopes which can be realized on orthogonal surfaces.
Finding Disjoint Paths on Directed Acyclic Graphs
"... Abstract. Given k + 1 pairs of vertices (s1, s2), (u1, v1),..., (uk, vk) of a directed acyclic graph, we show that a modified version of a data structure of Suurballe and Tarjan can output, for each pair (ul, vl) with 1 ≤ l ≤ k, a tuple (s1, t1, s2, t2) with {t1, t2} = {ul, vl} in constant time suc ..."
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Cited by 1 (0 self)
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Abstract. Given k + 1 pairs of vertices (s1, s2), (u1, v1),..., (uk, vk) of a directed acyclic graph, we show that a modified version of a data structure of Suurballe and Tarjan can output, for each pair (ul, vl) with 1 ≤ l ≤ k, a tuple (s1, t1, s2, t2) with {t1, t2} = {ul, vl} in constant time such that there are two disjoint paths p1, from s1 to t1, and p2, from s2 to t2, if such a tuple exists. Disjoint can mean vertex as well as edgedisjoint. As an application we show that the presented data structure can be used to improve the previous best known running time O(mn) for the so called 2disjoint paths problem on directed acyclic graphs to O(m log 2+m/n n + n log 3 n). In this problem, given four vertices s1, s2, t1, and t2, we want to construct two disjoint paths p1, from s1 to t1, and p2, from s2 to t2, if such paths exist. 1
Leftist Canonical Ordering
"... Abstract. Canonical ordering is an important tool in planar graph drawing and other applications. Although a lineartime algorithm to determine canonical orderings has been known for a while, it is rather complicated to understand and implement, and the output is not uniquely determined. We present ..."
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Abstract. Canonical ordering is an important tool in planar graph drawing and other applications. Although a lineartime algorithm to determine canonical orderings has been known for a while, it is rather complicated to understand and implement, and the output is not uniquely determined. We present a new approach that is simpler and more intuitive, and that computes a newly defined leftist canonical ordering of a triconnected graph which is a uniquely determined leftmost canonical ordering. 1
Incremental Convex Planarity Testing
, 2001
"... An important class of planar straightline drawings of graphs are convex drawings, in which all the faces are drawn as convex polygons. A planar graph is said to be convex planar if it admits a convex drawing. We give a new combinatorial characterization of convex planar graphs based on the decompos ..."
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An important class of planar straightline drawings of graphs are convex drawings, in which all the faces are drawn as convex polygons. A planar graph is said to be convex planar if it admits a convex drawing. We give a new combinatorial characterization of convex planar graphs based on the decomposition of a biconnected graph into its triconnected components. We then consider the problem of testing convex planarity in an incremental environment, where a biconnected planar graph is subject to online insertions of vertices and edges. We present a data structure for the online incremental convex planarity testing problem with the following performance, where n denotes the current number of vertices of the graph: (strictly) convex planarity testing takes O(1) worstcase time, insertion of vertices takes O(log n) worstcase time, insertion of edges takes O(log n) amortized time, and the space requirement of the data structure is O(n).